For the first 3 questions, state what type of problem it is



Test#3 Key:

1.) Given

U = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

A = { 1, 3, 5, 6, 7}

B = { 2, 3, 6, 7 }

C = { 4, 5 }

Find the following sets:

a.)

b.)

c.)

In problem #2 just set up, no need to compute the final answer.

2.) How many license plates consisting of 3 letters (from English alphabet)

followed by 2 digits are possible, if

a) repetitions are allowed.

263x102

b) repetitions are not allowed.

26x25x24x10x9=P(26,3)xP(10,2)

3.) In a survey of the students in a MAT119 class, it is found that

40 students are taking HIS100

45 students are taking ENG101

30 students are taking BUS301

5 students are taking all these three

10 students don’t take any of these three

15 students are taking both ENG101 and BUS301

10 students are taking both BUS301 and HIS100

25 students are taking both ENG101 and HIS100

a.) Draw a Venn diagram illustrating this problem and write down all relative numbers associated with this diagram.

b). How many students are there in this MAT119 class?

80

c). How many students are taking both HIS100 and ENG101, but not BUS301?

20

In problem #4 just set up, no need to compute the final answer.

4.) a.) How many different 5 letter words (real or not) can be formed from the first 8

letters of English alphabet, if A has to be the first letter in each of them and

no letters may be repeated?

P(7,4)=4x5x6x7

b.) In how many distinct ways can you plant 5 red tulips, 6 yellow daisies and 7

blue irises in a row in front of your house if you do not distinguish between

the flowers of the same kind.

C(18,5)xC(13,6)xC(7,7)= 18!/(5!6!7!)

5.) Suppose that you toss a fair coin 5 times.

a) How many possible outcomes do you have?

25

b.) How many outcomes have exactly 2 heads?

C(5,2)=10 (# of ways to place 2 heads in 5 positions, order not important since all heads look the same, then there is only 1 way to place all 3 tails in remaining 3 places)

c.) How many outcomes have at least 1 head?

# with 0 heads=1 (ttttt), so 25 - 1 = 31

In problem #6 just set up, no need to compute the final answer.

6.) Five cards are dealt randomly from a regular deck of 52 cards.

a) How many possible 5-card hands are there?

C(52,5) order not important

b) In how many ways can you select exactly 2 spades?

2 spades, rest not spades C(13,2)xC(39,3)

c) In how many ways can you select at most 2 red cards?

(2 or 1 or 0 red, remember that you select 5 each time)

C(26,2)xC(26,3) + C(26,1)xC(26,4) + C(26,0)xC(26,5)

7.) Experiment consists of rolling a balanced die twice.

a) List a sample space for that experiment.

36 outcomes S={(1,1), (1,2), ……. (1,6),

(2,1, (2,2), ……… (2,6),

……….

(6,1), (6,2), …….. (6,6)}

b) What is the probability that the first toss is at least 5

and the second toss is even.

(5,2), (5,4), (5,6)

(6,2), (6,4), (6,6) satisfy the event, call it A

P(A) = 6/36 = 1/6

c) Find the probability that the sum of the two tosses is not 11.

(5,6) and (6,5) have sum=11, so P(sum not 11) = 1 - 2/36 = 17/18

8.) The odds for John passing Math are 3 to 1, the odds against John passing

Biology are 7 to 5.

Let M=event that John passes Math B=event that John pasess Biology

a) Find the following probabilities:

b) Suppose that probability of John passing both Math and Biology

is 1/4, find the probability of John passing at least one of the two courses,

i.e. find :

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